Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle LOM = 7x - 77$, and $ m \angle MON = 5x - 73$, find $m\angle LOM$. $O$ $L$ $N$ $M$
From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 77} + {5x - 73} = {90}$ Combine like terms: $ 12x - 150 = 90$ Add $150$ to both sides: $ 12x = 240$ Divide both sides by $12$ to find $x$ $ x = 20$ Substitute $20$ for $x$ in the expression that was given for $m\angle LOM$ $ m\angle LOM = 7({20}) - 77$ Simplify: $ {m\angle LOM = 140 - 77}$ So ${m\angle LOM = 63}$.